If `3 tan ( theta -15^@) =- tan ( theta + 15^@) , ` then ` theta `=

Options

A. `n pi + pi/4`
B. `n pi + pi/8`
C. `n pi + pi/3`
D. none

Answer

To solve the equation \(3 \tan(\theta - 15^\circ) = -\tan(\theta + 15^\circ)\), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ 3 \tan(\theta - 15^\circ) = -\tan(\theta + 15^\circ) \]

### Step 2: Rearrange the equation
Rearranging gives:
\[ 3 \tan(\theta - 15^\circ) + \tan(\theta + 15^\circ) = 0 \]

### Step 3: Use the tangent addition formula
Using the tangent addition formula, we can express \(\tan(\theta - 15^\circ)\) and \(\tan(\theta + 15^\circ)\):
\[ \tan(\theta - 15^\circ) = \frac{\tan \theta - \tan 15^\circ}{1 + \tan \theta \tan 15^\circ} \]
\[ \tan(\theta + 15^\circ) = \frac{\tan \theta + \tan 15^\circ}{1 - \tan \theta \tan 15^\circ} \]

### Step 4: Substitute into the equation
Substituting these into the rearranged equation:
\[ 3 \left(\frac{\tan \theta - \tan 15^\circ}{1 + \tan \theta \tan 15^\circ}\right) + \left(\frac{\tan \theta + \tan 15^\circ}{1 - \tan \theta \tan 15^\circ}\right) = 0 \]

### Step 5: Clear the denominators
Multiply through by \((1 + \tan \theta \tan 15^\circ)(1 - \tan \theta \tan 15^\circ)\) to eliminate the denominators:
\[ 3(\tan \theta - \tan 15^\circ)(1 - \tan \theta \tan 15^\circ) + (\tan \theta + \tan 15^\circ)(1 + \tan \theta \tan 15^\circ) = 0 \]

### Step 6: Expand the equation
Expanding both sides:
\[ 3 \tan \theta - 3 \tan 15^\circ - 3 \tan \theta \tan^2 15^\circ + 3 \tan^2 15^\circ = \tan \theta + \tan 15^\circ + \tan \theta \tan^2 15^\circ + \tan^2 15^\circ = 0 \]

### Step 7: Collect like terms
Combine like terms:
\[ (3 \tan \theta - \tan \theta) + (-3 \tan 15^\circ - \tan 15^\circ) + (-3 \tan \theta \tan^2 15^\circ + \tan \theta \tan^2 15^\circ) + (3 \tan^2 15^\circ + \tan^2 15^\circ) = 0 \]
\[ 2 \tan \theta - 4 \tan 15^\circ - 2 \tan \theta \tan^2 15^\circ + 4 \tan^2 15^\circ = 0 \]

### Step 8: Factor out common terms
Factoring out \(2\):
\[ 2(\tan \theta - 2 \tan 15^\circ - \tan \theta \tan^2 15^\circ + 2 \tan^2 15^\circ) = 0 \]
This simplifies to:
\[ \tan \theta - 2 \tan 15^\circ - \tan \theta \tan^2 15^\circ + 2 \tan^2 15^\circ = 0 \]

### Step 9: Solve for \(\theta\)
Rearranging gives:
\[ \tan \theta (1 + \tan^2 15^\circ) = 2 \tan 15^\circ (1 + \tan^2 15^\circ) \]
Thus:
\[ \tan \theta = 2 \tan 15^\circ \]

### Step 10: Find general solution for \(\theta\)
Using the inverse tangent function:
\[ \theta = \tan^{-1}(2 \tan 15^\circ) + n\pi \]
This gives us the general solution for \(\theta\).

### Final Result
The value of \(\theta\) is:
\[ \theta = n\pi + \tan^{-1}(2 \tan 15^\circ) \]